| L(s) = 1 | + 3-s + 5-s − 5·7-s − 26·9-s + 10·11-s + 13·13-s + 15-s + 93·17-s − 82·19-s − 5·21-s + 192·23-s − 124·25-s − 53·27-s + 106·29-s − 172·31-s + 10·33-s − 5·35-s − 379·37-s + 13·39-s − 148·41-s − 329·43-s − 26·45-s + 631·47-s − 318·49-s + 93·51-s − 160·53-s + 10·55-s + ⋯ |
| L(s) = 1 | + 0.192·3-s + 0.0894·5-s − 0.269·7-s − 0.962·9-s + 0.274·11-s + 0.277·13-s + 0.0172·15-s + 1.32·17-s − 0.990·19-s − 0.0519·21-s + 1.74·23-s − 0.991·25-s − 0.377·27-s + 0.678·29-s − 0.996·31-s + 0.0527·33-s − 0.0241·35-s − 1.68·37-s + 0.0533·39-s − 0.563·41-s − 1.16·43-s − 0.0861·45-s + 1.95·47-s − 0.927·49-s + 0.255·51-s − 0.414·53-s + 0.0245·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - p T \) |
| good | 3 | \( 1 - T + p^{3} T^{2} \) |
| 5 | \( 1 - T + p^{3} T^{2} \) |
| 7 | \( 1 + 5 T + p^{3} T^{2} \) |
| 11 | \( 1 - 10 T + p^{3} T^{2} \) |
| 17 | \( 1 - 93 T + p^{3} T^{2} \) |
| 19 | \( 1 + 82 T + p^{3} T^{2} \) |
| 23 | \( 1 - 192 T + p^{3} T^{2} \) |
| 29 | \( 1 - 106 T + p^{3} T^{2} \) |
| 31 | \( 1 + 172 T + p^{3} T^{2} \) |
| 37 | \( 1 + 379 T + p^{3} T^{2} \) |
| 41 | \( 1 + 148 T + p^{3} T^{2} \) |
| 43 | \( 1 + 329 T + p^{3} T^{2} \) |
| 47 | \( 1 - 631 T + p^{3} T^{2} \) |
| 53 | \( 1 + 160 T + p^{3} T^{2} \) |
| 59 | \( 1 + 478 T + p^{3} T^{2} \) |
| 61 | \( 1 + 300 T + p^{3} T^{2} \) |
| 67 | \( 1 + 722 T + p^{3} T^{2} \) |
| 71 | \( 1 + 335 T + p^{3} T^{2} \) |
| 73 | \( 1 - 90 T + p^{3} T^{2} \) |
| 79 | \( 1 - 788 T + p^{3} T^{2} \) |
| 83 | \( 1 - 96 T + p^{3} T^{2} \) |
| 89 | \( 1 + 866 T + p^{3} T^{2} \) |
| 97 | \( 1 + 998 T + p^{3} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.235288348421525402610575045348, −8.679873718765186057468371877457, −7.76592227579020819921510052824, −6.76878570619522263159046273302, −5.86449024564815969138856734514, −5.04488441397912295017881277607, −3.67541042955256492568459758768, −2.91481749222069522473013106574, −1.52773470219585414266868626172, 0,
1.52773470219585414266868626172, 2.91481749222069522473013106574, 3.67541042955256492568459758768, 5.04488441397912295017881277607, 5.86449024564815969138856734514, 6.76878570619522263159046273302, 7.76592227579020819921510052824, 8.679873718765186057468371877457, 9.235288348421525402610575045348
